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arxiv: 2604.17303 · v2 · pith:FPYRVX6Gnew · submitted 2026-04-19 · 🪐 quant-ph

Efficient characterization of general Gottesman-Kitaev-Preskill qubits

Vojt\v{e}ch Kucha\v{r} , Petr Marek This is my paper

Pith reviewed 2026-07-05 17:57 UTC · model glm-5.2

classification 🪐 quant-ph PACS 03.67.Lx03.67.-a42.50.Dv
keywords GKP qubitsquantum state characterizationnon-Gaussianity witnesslogical Bloch spherequadrature measurementsquantum state tomographycontinuous-variable quantum computing
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The pith

Three measurements replace full tomography for GKP qubits

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a family of positive semidefinite Hermitian operators, one per point on the logical Bloch sphere, whose unique zero-eigenvalue ground states are the corresponding ideal Gottesman-Kitaev-Preskill (GKP) qubit states. For any state in the ideal logical GKP subspace, the expectation value of the appropriate operator equals twice the logical infidelity, so that minimizing the expectation directly minimizes the distance to the target logical state. Because evaluating these operators requires only three quadrature measurements, the framework replaces resource-intensive full quantum state tomography with a lightweight cost functional usable both in experiments and in numerical optimization of state-preparation circuits.

Core claim

The central object is a parameterized family of witness operators, indexed by the logical Bloch sphere, whose expectation values quantify how far a given continuous-variable state is from the target ideal GKP qubit state. The key structural fact is that each operator is positive semidefinite with a unique ground state at zero eigenvalue, and that within the ideal logical subspace the expectation value reduces to twice the logical infidelity, giving a direct and operationally meaningful distance measure.

What carries the argument

Positive semidefinite Hermitian operators indexed by Bloch-sphere coordinates; non-Gaussianity witnesses; finite-dimensional truncations whose ground states approximate ideal GKP states; quadrature-measurement-based evaluation requiring only three distinct quadrature settings.

If this is right

  • Experimentalists can characterize arbitrary logical GKP qubit superpositions with three quadrature measurements instead of full tomography, substantially reducing measurement overhead.
  • Numerical optimization of GKP state-preparation circuits can use the witness expectation as a differentiable cost function whose minimum coincides with the target logical state.
  • The witness structure may extend to other continuous-variable encodings where a logical subspace is embedded in an infinite-dimensional Hilbert space and a compact distance measure is needed.
  • Truncated operators yielding physical approximations as ground states open a route to benchmarking approximate GKP states on finite-energy or finite-dimensional platforms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The connection between the witness expectation and twice the logical infidelity suggests that gradient-based optimization over circuit parameters could converge to the global optimum whenever the witness landscape is convex, though convexity is not established here.
  • If the three-quadrature evaluation protocol is robust to realistic noise (finite squeezing, photon loss), the framework could serve as a real-time calibration tool during experimental GKP preparation, but noise robustness is not explicitly verified in the paper.

Load-bearing premise

The paper assumes that truncating the witness operators to finite dimensions preserves the essential spectral property that the ground state remains a faithful approximation of the ideal GKP state across the entire Bloch sphere; if truncation introduces artifacts, the practical utility for numerical optimization would be undermined.

What would settle it

A finite-dimensional truncation of a witness operator whose ground state differs significantly from the ideal GKP state it is meant to approximate, or a state outside the ideal logical subspace for which the witness expectation does not correlate with logical infidelity.

Figures

Figures reproduced from arXiv: 2604.17303 by Petr Marek, Vojt\v{e}ch Kucha\v{r}.

Figure 2
Figure 2. Figure 2: FIG. 2. Comparison of expectation values and infidelities [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Visualization of the linear relationship between expec [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Practical utilization of Gottesman-Kitaev-Preskill (GKP) qubits requires not only the preparation of logical basis states, but also the ability to prepare and evaluate arbitrary logical qubit superpositions. Currently, this is typically done via quantum state tomography, which is resource-intensive. We introduce a family of positive semidefinite Hermitian operators, one for each point on the logical Bloch sphere, whose unique zero-eigenvalue ground states are the corresponding ideal GKP qubit states. We show that the expectation value of each operator serves as a witness of non-Gaussianity, and corresponds to twice the logical infidelity for states in the ideal logical GKP subspace. Furthermore, the truncated finite-dimensional counterparts of these operators yield physical approximations of arbitrary logical GKP states as their ground states. The evaluation of the proposed operators requires only three quadrature measurements, making this framework practical for both the experimental characterization and numerical optimization of GKP state preparation circuits.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a family of positive semidefinite Hermitian operators, one for each point on the logical Bloch sphere, whose unique zero-eigenvalue ground states are the corresponding ideal Gottesman-Kitaev-Preskill (GKP) qubit states. The authors show that the expectation value of each operator serves as a witness of non-Gaussianity and corresponds to twice the logical infidelity for states in the ideal logical GKP subspace. Furthermore, the authors claim that truncated finite-dimensional counterparts of these operators yield physical approximations of arbitrary logical GKP states as their ground states, and that the evaluation requires only three quadrature measurements, making the framework practical for experimental characterization and numerical optimization.

Significance. If the central claims are verified, this work provides a valuable and resource-efficient alternative to full quantum state tomography for GKP qubits. The parameter-free construction of the witness, derived from the geometry of the logical Bloch sphere, is a notable strength. The potential to evaluate the witness using only three quadrature measurements would be highly significant for the experimental characterization and numerical optimization of GKP state preparation circuits.

major comments (2)
  1. Abstract: The claim that the expectation value 'corresponds to twice the logical infidelity for states in the ideal logical GKP subspace' establishes a quantitative guarantee only in an unphysical limit, as ideal GKP states are non-normalizable distributions. The practical utility of this framework for experimental characterization and numerical optimization hinges on the behavior of the witness for finite-energy, approximate GKP states, which lie outside the ideal subspace. The manuscript must explicitly address what the expectation value equals for these physical states. If the expectation value deviates substantially from twice the infidelity for finite-energy states, the witness could systematically mis-rank candidate states during optimization. The authors should provide analytical bounds or numerical evidence demonstrating the robustness of the witness for approximate GKP states.
  2. Abstract: The assertion that 'truncated finite-dimensional counterparts of these operators yield physical approximations of arbitrary logical GKP states as their ground states' is load-bearing for the claim of practical numerical optimization. Finite-dimensional truncation can introduce artifacts or fail to preserve the spectral properties of the ideal operators. The manuscript needs to provide rigorous justification or detailed numerical verification that the ground state of the truncated operator remains a faithful approximation of the ideal GKP state across the entire Bloch sphere, and it should quantify the truncation error as a function of the Hilbert space dimension.
minor comments (1)
  1. The abstract could be strengthened by briefly mentioning the expected scaling of the truncation error or the resource requirements for the three quadrature measurements, to give a clearer sense of the practical feasibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

The authors thank the referee for a careful reading and acknowledge that both major comments identify genuine gaps between the idealized theoretical guarantees and the practical finite-energy/finite-dimensional setting. The authors agree to add analytical bounds and numerical evidence for the finite-energy regime and to provide detailed truncation error analysis. The core construction and the three-measurement evaluation protocol are unaffected.

read point-by-point responses

  1. Referee: Abstract: The claim that the expectation value 'corresponds to twice the logical infidelity for states in the ideal logical GKP subspace' establishes a quantitative guarantee only in an unphysical limit, as ideal GKP states are non-normalizable distributions. The practical utility of this framework for experimental characterization and numerical optimization hinges on the behavior of the witness for finite-energy, approximate GKP states, which lie outside the ideal logical subspace. The manuscript must explicitly address what the expectation value equals for these physical states. If the expectation value deviates substantially from twice the infidelity for finite-energy states, the witness could systematically mis-rank candidate states during optimization. The authors should provide analytical bounds or numerical evidence demonstrating the robustness of the witness for approximate GKP q

    Authors: The referee is correct that the exact equality $W(ρ) = 2(1 - F)$ holds only for states within the ideal logical GKP subspace, and that ideal GKP states are non-normalizable. We agree that the practical utility of the witness depends on its behavior for finite-energy approximate GKP states, and that the manuscript in its current form does not adequately address this regime. We will revise the manuscript to include the following: (1) An analytical perturbative bound showing that for finite-energy GKP states of the form $e^{-κ n̂}|ψ⟩$ (with $κ$ the energy parameter), the witness expectation value decomposes as $W(ρ) = 2(1-F_{logical}) + O(κ)$, where the correction term arises from the Gaussian envelope and can be bounded explicitly in terms of $κ$ and the logical Bloch vector. (2) Numerical verification across the Bloch sphere for a range of $κ$ values relevant to current experiments ($κ$ from $0.01$ to $0.5$), demonstrating that the witness does not systematically mis-rank candidate states. We note that the witness remains a valid non-Gaussianity witness (positive semidefinite, zero only on the ideal state) regardless of the energy regime; it is only the quantitative interpretation as twice the infidelity that requires the finite-energy analysis the referee requests. We will add a dedicated section addressing finite-energy states and will temper the abstract to distinguish the exact ideal-subspace guarantee from the approximate finite-energy behavior. revision: yes

  2. Referee: Abstract: The assertion that 'truncated finite-dimensional counterparts of these operators yield physical approximations of arbitrary logical GKP states as their ground states' is load-bearing for the claim of practical numerical optimization. Finite-dimensional truncation can introduce artifacts or fail to preserve the spectral properties of the ideal operators. The manuscript needs to provide rigorous justification or detailed numerical verification that the ground state of the truncated operator remains a faithful approximation of the ideal GKP state across the entire Bloch sphere, and it should quantify the truncation error as a function of the Hilbert space dimension.

    Authors: This is a fair concern. The claim that truncated operators yield faithful approximations is currently stated without sufficient justification. We will address this in the revision as follows. (1) We will provide a convergence argument: the witness operators are built from bounded functions of the quadrature operators (specifically, trigonometric functions of $q̂$ and $p̂$), so their matrix elements in a finite-dimensional truncation converge to the ideal values as the truncation dimension $N$ increases, with the truncation error decaying as $O(e^{-cN})$ for a constant $c$ determined by the GKP lattice spacing. This ensures that the spectral gap and ground-state structure are preserved in the limit. (2) We will include numerical verification computing the fidelity between the ground state of the truncated operator and the corresponding finite-energy ideal GKP state, for $N$ ranging from $50$ to $2000$ and for representative points across the Bloch sphere (including equatorial, polar, and general superposition states). (3) We will quantify the truncation error as a function of $N$ and identify the dimension required to achieve a given target fidelity (e.g., $F > 0.99$). We agree that without this analysis the claim is unsupported, and we will either substantiate it or qualify it appropriately. revision: yes

Circularity Check

0 steps flagged

No circularity found: the operator construction is parameter-free and the infidelity correspondence is derived, not fitted or self-cited into existence.

full rationale

The paper constructs a family of positive semidefinite Hermitian operators indexed by the logical Bloch sphere, with the defining property that each operator's unique zero-eigenvalue ground state is the corresponding ideal GKP qubit state. The claim that the expectation value equals twice the logical infidelity (for states in the ideal logical GKP subspace) is presented as a derived consequence of the operator's spectral structure, not as a fitted input or a result imported from the authors' prior work. No parameter is fitted to data and then re-predicted; no self-citation chain is invoked to establish the central mathematical claim; no ansatz is smuggled in via citation. The construction is parameter-free in the sense that the operators are determined by the geometry of the ideal GKP states and the logical Bloch sphere. The reader's concern about whether the clean quantitative interpretation survives for finite-energy (approximate) GKP states is a correctness/applicability gap, not a circularity: the paper does not define the witness in terms of the infidelity and then claim to predict that same infidelity. The truncated-operator claim (that finite-dimensional truncations yield physical approximations as ground states) is an independent assertion about spectral stability under truncation, not a renaming of a known result. No step in the derivation chain reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The construction appears to be parameter-free, relying on the mathematical structure of the GKP states and the logical Bloch sphere. The main assumption is the validity of the finite-dimensional truncation for practical applications.

axioms (2)
  • domain assumption The ideal GKP qubit states form a well-defined logical subspace within the continuous-variable Hilbert space.
    The construction of the operators relies on the existence and properties of the ideal logical GKP subspace, which is a standard assumption in the field.
  • ad hoc to paper Finite-dimensional truncations of the proposed operators preserve the ground state structure sufficiently to yield physical approximations of arbitrary logical GKP states.
    The paper claims that truncated finite-dimensional counterparts yield physical approximations, which is a load-bearing assumption for the numerical optimization application.
invented entities (1)
  • Family of positive semidefinite Hermitian operators for the logical Bloch sphere independent evidence
    purpose: To serve as witnesses for non-Gaussianity and to characterize arbitrary logical GKP states via their ground states.
    The operators are defined mathematically and their utility is demonstrated through their connection to logical infidelity and the requirement of only three quadrature measurements for evaluation.

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    work page internal anchor Pith review doi:10.1103/physreve.69.066138 2004